Introduction^ This paper originated in trying to show that the one point compactification of an orientable generalized ^-manifold (n-grn) with cohomology ...
THE END POINT COMPACTIFICATION OF MANIFOLDS FRANK RAYMOND
Introduction^ This paper originated in trying to show that the one point compactification of an orientable generalized ^-manifold (n-grn) with cohomology isomorphic to Euclidean n-space was an orientable n-gm. Heretofore, in papers on transformation groups where this was relevant, it was stated as an extra assumption.1 The solution to this problem is given as a corollary to the main theorems which characterize the orientable (or locally orientable) generalized manifolds whose Freudenthal end point compactification is again an orientable (or locally orientable) generalized ^-manifold (see 4.5 and 4.13). In the first section we give a new characterization of the Freudenthal end point compactification in terms of inverse limits. We show as in Specker [10] that a certain O-dimensional cohomology group measures the extent of this compactification. Higher dimensional analogues of this cohomology group have been used by Conner [4] in proving, e.g., that a simply connected locally Euclidean ^-manifold whose 1 point compactification is a locally Euclidean w-manifold cannot be fibered by a non-trivial compact fiber. He has called these groups the cohomology of the ideal boundary. These groups are further studied and the homology analogue is derived (see § 2). The main Lemma (2.16) is an exact sequence which relates the homology of the ideal boundary with the homology of a given compactification and the local homology groups at infinity. This exact sequence together with our characterization of the Freudenthal compactification gives the main theorems. Applications are given in § 3 to Poincare duality, in § 5 to open 2manifolds, and in § 6 special mappings of manifolds. Throughout this paper X will denote a locally campct, locally connected, connected Hausdorff space. If S is a locally compact Hausdorff space, A(S) will denote the collection of open subsets of S whose closure is compact. When the generic space is not necessarily locally connected, as is usually the case in § 2 and part of § 4, it will be denoted by the letter S. I. The Freudenthal end point compacification* 1.1. LEMMA. If V, U e A(X) such that V a U, then at most a finite number of components of X — V meet in X — U. In particular, Received July 14, 1959. National Science Foundation Fellow. ] See, for example, Montgomery and Mostow, Toroid transformation groups on Euclidean space, 111. J. of Math. 2 (1958), or [14]. 947
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FRANK RAYMOND
for every V e A(X) there is at most a finite number of unbounded2 components of X — V. 1.2. DEFINITION (Specker). The space X has at least k ends if there exists V e A(X) such that X — V has at least k unbounded components. The space X has exactly k ends if X has at least k ends but not at least k + 1 ends. 1.3. If V, U e A(X), F c 17, then each unbounded component of X — U lies in an unbounded component X — V. Furthermore, each unbounded component of X — V meets at least one unbounded component of X — U. If X has exactly k ends then there exists V e A(X) so that X ~ V has exaclty k unbounded components. Thus if F c ί ί e A(X)y U would have the same property. Let us therefore in case X has exactly k ends, associate an ideal point with each unbounded component of X — V. There is a one-to-one correspondence of the unbounded components of X — U with those of X — V as we have just seen. We may adjoion this ideal set of k points to X and specify that a neighborhond of such an ideal point will be the ideal point together with the unbounded component associated to the ideal point. Such a process yields a compactification of X by k points. It is exactly this process that we wish to extend in case X has more than a finite number of ends. 1.4. Let A index the set A(X). Partially order A by a < β if VΛ, Vβ e A(X) and Va c Vβ. Let Aa = {Aί}t«=f( Aa be the mapping induced from the inclusion (X— Vβ) c (X — Va), i.e., πβΛ sends each unbounded component Aιβ of X — Vβ into the unique unbounded component of X — F α which contains Ajg. Clearly, ττ£ is owto. T/ie collection {Aω, π%\ forms an inverse system of sets. Let B = inverse limit {Aα, π%\. Each set AΛ is a finite set and if topologized by the discrete topology the space B will be a compact space. A topology will now be put on the set X' = X U B. Let a e B, and define a neighborhood system for the point a. Let a e A. Let πa(a) = A*(α) be the undounded component of Aα which is the αth coordinate of a. Let B«{a) be the set of points of B so that πΛ{b) = A£(α). {a) (α) Then Λ/^ = A% U £* will be an element of the neighborhood system r of a e B c X . The neighborhood system of a will be {iV?}*^. If f x e X a X & neighborhood system of x will be given by just choosing a neighborhood system of x in X. 1.5. 2
THEOREM.
Let Xf be the space obtained from
A set is unbounded if its closure is not compact.
X as defined
THE END POINT COMPACTIFICATION OF MANIFOLDS
above.
949
Then,
(a)
X ' is Hausdorjf, connected and locally connected,
(b)
B is closed and has no interior
points in X},
(c) B is totally disconnected in X', (d) the topology of B as an inverse limit
is the relative
topology
of B in X', (e)
X' is compact,
(f)
if 0 is an open connected set in X', then O — B is connected.
Proof. The definition of Xf together with the fact that every point of B is a limit point of X implies (a) and (b). Let a,b e B, a Φ b. Then there exists a e A such that πa{a) Φ πa{b). Let C = B - BMa). Clearly b e C and 5* ( α ) is open in B (not in Xr). Now, πjjp) = πa(a) if p is a limit point of B«{a), hence B«{a) is both open and closed. Consequently, a and b are separated in B, hence (c) holds. Let a e B and N% be an element of the neighborhood system of α. The set BΛ{a) consists of all those points of b e B so that πa(b) = A%{a) = Tijia). However, in terms of the inverse limit topology on β, Ba{a) = π~1(A%{a)) is also an open set. Moreover, the neighborhoods of a e B in terms of the inverse limit topology on B are generated precisely by the sets B*ia). Hence, (d) holds and, therefore, B is compact. It is now not hard to see that Xf must be compact. Let 0 be any open connected set which meets B. The set 0 may be regarded as the union of fundamental open connected sets of points of 0 Π B. Each such neighborhood is connected and cannot be separated by removal of B*ia), by definition of neighborhoods. An elementary argument now yields that O — B must be connected. In particular, every point of B is a local non-cut point of X''. 1.6. DEFINITION. The compactification X' of X will be called the (Freudenthal) end point compactification of X. 1.7. The set B contains exactly k points, if and only if, X has exactly k ends. In case k is finite, Xr agrees with the preliminary definition of the end point compactification of 1.3. 1.8. The consequences of Theorem 1.5 serve as an abstract characterization of the end point compactification. THEOREM.
Let X* be a compactification of X such that:
(a)
X * is connected;
(b)
X is open is X*;
(c)
X * — X is totally disconnected;
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FRANK RAYMOND
(d) if p e X* — X and U is a connected neighborhood (open) of p then U — (X* — X) is connected. f Then, X* may be identified with the end point compactification X of X. Proof. As X* is to be locally compact, it can fail to be locally connected only on a set containing a non-degenerate continuum [see e.g., 12, 2.2, pp. 104]. But X is locally connected and B* = X* - X is totally disconnected, and therefore, X* must be locally connected. Moreover, X* — X has no interior points in X*. Let p be a point of S*. Let Up be a neighborhood such that Up - B* is unbounded in X. Let 5* - Up = C. Then as B* is totally disconnected C and p may be separated. Choose C1 and C2 such that d U C2 = £*, p e Clf Cd C2, C . n C ^ φ. Since the C, are both closed and disjoint we can choose open disjoint sets O1 and O2 such that Ox D C19 and O2 Z) C2. We can choose both 0* to be regular open sets (i.e., interior Ot — 0*), and Ox c Up. As Ox and O2 cover B* we may select from O1 and O2 just those components which meet B*. Let U = be the components of Ox and O2 which meet J5*. As U is chosen to be a regular open set, X* — U = F for some F e A(X). The collection A'(X) of all possible F of this form is cofinal in A(X). Now by taking the unbounded (in X) components of X* — F we can recover, by the process used in defining X', a cofinal neighborhood system of p. Thus it readily follows that 2?* is homeomorphic to the inverse limit of the inverse system of unbounded components of the complement of the closure of elements of A'(X). Thus X* may be identified with X' and £* with B. 1.9. The abstract characterization of the end point compactification used the same conditions as that of Freudenthal [7] although the postulates on the topology of X are not quite the same. 1.10. In the case of a complex, Specker has measured the number of ends by the rank of a certain cohomology group, [10]. Analogously, the similar cohomology group, but now of the space X, gives the cohomology of B (see 1.13). Let H*(X) denote the Cech cohomology ring of X with coefficients in the principal ideal domain L. Then following Conner [4] define 7° (JSΓ) to be the direct limit of H°(X - F) = direct limit of H°(X - V), taken over the directed set A, V e A(X) The group H°(X — F) splits into the direct product of copies of L, one for each component. In the direct limit every contribution from an unbounded component is annihilated. In fact, if X is paracompact it is easy to show that 7°(X) is free and at most countably generated. 1.11.
Let Xf denote the end point compactificatoin of X, X' — X =
THE END POINT COMPACTIFICATION OF MANIFOLDS
B, and V e A(X). 1.12
Consider the exact sequence: r
H\X
0
951
-
V,X-
H\X'
~^
-
H\X - V)
H\X' -V,X-
V)
The group H°(X' - F, X - V) & H\X', X) = 0, as X' is connected. The mapping i* : H°(X' — V) —>H°(X— V) is an isomorphism when restricted to the unbounded components (in X) of X' — V. Furtherf more, the direct limit H°(X - V) = direct limit if°(X' - V), and by continuity is isomorphic to H°(B). Passing the direct limit of 1.12 we obtain: 1.13. THEOREM. The mapping it : H°(B) ι phism onto. Moreovery H {Xs; X' — B) is 0.
I°(X) is an isomor-
2 Cohotnology and hotnology of the ideal boundary. In this section we shall recall the definition of end groups or the cohomology groups of the ideal boundary as given by Conner [4]. The higher dimensional analogue of (1.12) will be developed (2.6) and an analogous definition for homology of the ideal boundary will also be given. In § 3 we turn to applications. 2.1. Let C*(S) denote the grating of Alexander-Spanier cochains of the locally compact Hausdorff space S. The coefficient domain, usually suppressed, is taken to be a principal ideal domain, L. Let C*(S) denote the subgrating of elements of C*(S) which have compact support; C*(S) is an ideal in C*(S). The quotient ring is called the cohomology ring of the ideal boundary of S, or the end cohomology ring of S. It will be denoted, following Conner [4] by I*(S) = Σ£=oIp(S). If / is a proper mapping of S into T then the daigram
HΊ(S)
j/*
(2.2)
H'(S)
/*
ΐ/*
H>(T) is commutative. That the horizontal rows are exact is a consequence of the exactness of the sequence of cochains: (2.3)
0
C&S) -?-> C'(S) —^ C»(S)IC»(S)
>0
and the fact that i and j commute with the coboundary. Let U e A(S), then it is to be remarked that (2.3) is the direct limit of: (2.4)
0
Cl( U)
> C'(S)
> C'(S)IC*( U)
>0.
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FRANK RAYMOND
The cohomology functor commutes with the direct limit and therefore the top horizontal line of (2.2) may be written as the direct limit of derived cohomology sequence of (2.4), i.e. the direct limit of the exact sequence: A*
(2.5)
A*
A*
> H%U) -^-> H*(S) -^-> H*(S - U)-^-» .
Let C*(S, U) denote the subgrating of C*(S)' of elements whose support dose not meet U. The derived cohomology ring is denoted as usual by H*(S, U). The quotient grating, CI(U) = C*(S)IC*(S, U) has derived cohomology ring isomorphic to H*{U). f Let S be an open subset of a compact space S . Let B = S' — S and U e A(S). The sequence; f
> H*>(S - U,S-
p
U)-±-+ H (S' - U) —
H*(S - S)
—
is exact and the direct limit may be identified with (2.6)
> Hp(S', S) —
H*(B) —
Ip(S) -^U H»+1(S', S) .
2.7. In the case of a polyhedral manifold Sf with manifold boundary B, H*(S',S) = 0. Thus, i* : H*(B) — I*(S) is a ring isomorphism. This could be used as a justification of the term "cohomology of the ideal boundary''. 2.8. In developing the analogous concept of end homology groups it is necessary to follow the definition of Borel and Moore for the homology of a locally compact Hausdorff space S [14]. As these seminar notes [14] are not yet readily available, we shall recall the definitions and some of the porperties of the homology groups. These homology groups are determined (2.8, (2)) by the Alexander-Spanier cohomology groups with compact support. They are used by Borel and Moore to obtain the Poincare duality for generalized manifolds with an arbitrary principal ideal domain as coefficient domain, and in terms of a homology computable in terms of the cohomology. If L is a field, the homology groups are isomorphic to the single space Cech homology groups of [6]; if L is the integers then they may differ from the single space Cech homology groups. Let 0 —> L —> AQ —• Aλ —> 0 be an injective resolution of L, (L is hereditary). This permits one to regard A = Ao + Aλ as an augmented complex over L. Let C = Hom(Cc*(S), A). Define CP(S) = Horn (C*(S), A) 0 Horn (C?+1(S), Ax) with differential df(c) = d(f(c)) + +I ( - 1)* f(dc), f e Cp, c e C*(S). It is shown in [14] that: (1) C*(JS) is a complete grating (2) for each open U c S, the seqμence
THE END POINT COMPACTίFICATION OF MANIFOLDS
953
+1
0
> Ext (H* (C*(U))f L) > HP (Horn (Cϊ(U), A) P >Ή.om (H (C*(U)), L) > 0, is exact and splits. (3) The above exact sequence is compatible with homomorphisms. (4) The associated sheaf is flabby (flasque) (or φ — fine, if φ is a paracompactifying family). 2.9. The homology group HP(S;L(L to be Hp(Hom(C*(S),A)).
usually suppressed) is defined
2.10. If U is an open subset of S then C*(U), and C*(S)IC*(U) can be used to define the homology groups H*(U; L) and H*(S — U; L) respectively, [14]. We obtain the exact sequence: (2.11)
> HP(S - tf)-^U HP(S) —
HP{U) -^U H'-^S - U)
>
as the derived homology sequence of exact (A is injective) sequence: (2.12)
0
> Horn (C*(S)IC*(U), A) — > Horn (C*(S), A) > Horn (C*(U), A)
>0.
C
2.13. The group H P(S) will denote the pth homology of those chains CP(S) of CP(S) with compact support. (The support of a chain is defined analogously to that in [1], or equivalently, as the support of the cross section which represents the given chain in the associated sheaf of chains.) The subgrating Cl(S) may be identified with the direct limit of CP(V) = Horn (C*(S)/C*(S - V), A) taken over all Ve A(S). This follows from the fact that the map Horn (CC*(S)/CC*(S - M), A) — Horn (Cc(S)ICt(S — N), A) is an injection for all compact M, N, M c N c S. Thus, HP(S) may be identified with the direct limit of HP(M), for all compact M c S. We shall define H;(S, U) to be HP(C*(S)IQ(U)). This is identified, therefore, with the direct limit, over all compact M c U, of HP(S - M). As the sequence: (2.14)
0
> C^S) -*U C , ; ( S ) i C*(S)IC%(S)
>0
is exact and the boundary operator decreases the supports we can obtain the derived exact sequence: (2.15)
> H;(S) JU
HP(S) -iίU Ip(S)-^> HUS)
>
Clearly, IP(S) = Hp(C*(S)ICi{S)) identifies itself with the direct limit over the compact sets M c S of HP(S — M). In particular, IP(S) = direct limit over V e A(S) of HP(S ~ V). For a locally finite polyhedron S, I^(S) is isomorphic to the homology of the infinite chains mod the finite chains. (The term "homology of the ideal boundary'' for I*(S) may not be justified as in the case for cohomology
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FRANK RAYMOND f
because if S were a polyhedral manifold with boundary B, S = S' — B, then IP+1(S) = HP(B), (cf. 2.7). That is, one might expect from the name we have chosen, that IP(S) = HP{B). But the decision to call I*(S) the homology of the ideal boundary of S rests on the similarity, on the chain level, of the definitions of cohomology of the ideal boundary and the results of § 3.) 2.16. THEOREM. Let S c S' as an open subset of the locally compact Hausdorff space S'. Then the sequence: > HP(S' - S) —
Hl(S', S) —
Ip(S)—
HP^(S' -S)
>
is exact. Proof. Let M be a compact set c S. Then consider the exact sequence of chains: (2.17)
0
> CP(S' - S)
> CP{S' - M)
> CP(S - M)
>0
Passing to the direct limit over all compact M a S oί the derived homology sequence of (2.17) we obtain by use of the identification discussed above the desired result. 2.18. Observe that if Sf denotes the one point compactification of S, o\ : HftS', S) -> IP(S) is an isomorphism, p > 0. The group HCP(S', S* in this case is precisely the local homology group Sf at oo (oo = S' — S)) 3 Poincare duality for the ideal boundary. We shall adhere to the terminology of [9] and [1] for generalized ^-manifolds (w-gm's). The ring of coefficients is still the principal ideal domain, L. 3.1. THEOREM. the diagram e
> H p(X)
Let X be a paracompact
orientable n-gm.
Then,
>1
—> Hnp(X)
> In~\X)
> HTP
is commutative. The horizontal rows are exact and the vertical are isomorphisms (Poincare duality).
maps
Proof. This may be obtained from the five lemma by just defining a natural map from CP(X)/CCP(X) to Cn~p(X)ICΓp(X) by means of a map induced from Poincare duality for orientable w-gm's. with compact and closed support, respectively. An alternate procedure would be to let U e A(X) and to consider the commutative diagram
THE END POINT COMPACTIFICATION OF MANIFOLDS
H;(U)
II
> H,(X)
> HV{X, U)
n
n
—> H -»(X)
I
> H -"(X-
-U
9§5
Hv
. i
U) -^->
Hrp+1(U)
The group HP(X, U) is computed by taking the homology of the chains of X mod those chains of X whose support (closed in X, hence, compact in U) lie in U. Each vertical map is an isomorphism. (This explicit form of the Poincare duality theorem for homology manifolds, discussed in detail in the author's dissertation, University of Michigan, 1958, is derived from [2; Ch. 19, Th. 7].) Now by passing to the direct limit over all U e A(X) we obtain the desired reslut. A Poincare duality theorem for non-orientable manifolds may be obtained if twisted coefficients are used. 4* The end point compactification of generalized manifolds. 4.1. Let S be an open subset of a compact Hausdorff space S'. Let B = S' — H9(B)
> HCP(S', S)
> I,(S)
H
which splits into the direct product of the exact sequences: (d)
> HP{B,)
C
> H P(S', S' - Bt)
> IP(S' - B
